Applying Dynamical Systems Theory to Optimize Libration Point Orbit Stationkeeping Maneuvers for WIND

Abstract

NASA's WIND mission has been operating in a large amplitude Lissajous orbit in the vicinity of the interior libration point of the Sun-Earth/Moon system since 2004. Regular stationkeeping maneuvers are required to maintain the orbit due to the instability around the collinear libration points. Historically these stationkeeping maneuvers have been performed by applying an incremental change in velocity, or Δv, along the spacecraft-Sun vector as projected into the ecliptic plane. Previous studies have shown that the magnitude of libration point stationkeeping maneuvers can be minimized by applying the Δv in the direction of the local stable manifold found using dynamical systems theory. This paper presents the analysis of this new maneuver strategy which shows that the magnitude of stationkeeping maneuvers can be decreased by 5 to 25 percent, depending on the location in the orbit where the maneuver is performed. The implementation of the optimized maneuver method into operations is discussed and results are presented for the first two optimized stationkeeping maneuvers executed by WIND. Nomenclature A(t)= partial derivative matrix of the CR3BP equations of motion with respect to the six element state b = y-intercept for a line = vector from larger primary, m 1 , to the spacecraft, m 3 DF(X) = partial derivative matrix of the free variables, X, with respect to the constraints, F(X) d = scalar distance from the larger primary, m 1 , to the spacecraft, m 3 ̅ = nondimensional vector from the larger primary, m 1 , to the spacecraft, m 3 D 1 = distance from m 1 ,m 2 barycenter to larger primary, m 1 D 2 = distance from m 1 ,m 2 barycenter to smaller primary, m 2 F(X) = constraint equations for fixed-time multiple shooting algorithm ̅ = equations of motion for the spacecraft in the CR3BP = dimensional gravitational constant l * = characteristic length m * = characteristic mass m 1 = mass of the larger primary m 2 = mass of the smaller primary m 3 = mass of the spacecraft (zero) n = number of patch points ̅ = dimensional position vector of m 3 with respect to the system barycenter = vector from smaller primary, m 2 , to the spacecraft, m 3 r = scalar distance from the smaller primary, m 2 , to the spacecraft, m 3 ̅ = nondimensional vector from the smaller primary, m 2 , to the spacecraft, m 3 s = slope of a line t = time 2 Figure 1. WIND Spacecraft and Instruments 1 . t * = characteristic time U * = pseudo-potential function X = free variables vector for fixed-time multiple shooting algorithm x = x-component of position vector x t = end points of an integrated state after time t y = y-component of position vector z = z-component of position vector α = maneuver direction Δv= maneuver magnitude θ = orientation of rotating frame relative to inertial frame μ = nondimensional mass ratio ̅ = nondimensional position vector of m 3 with respect to the system barycenter τ = nondimensional time φ = state transition matrix = differentiation with respect to time

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